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Strategies for increasing the Néel temperature of magnetoelectric Fe2TeO6
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 022203 (http://iopscience.iop.org/0953-8984/27/2/022203) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 155.246.15.35 This content was downloaded on 21/02/2015 at 11:26
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 022203 (5pp)
doi:10.1088/0953-8984/27/2/022203
Fast Track Communication
´ Strategies for increasing the Neel temperature of magnetoelectric Fe2TeO6 Sai Mu and Kirill D Belashchenko Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, NE 68588, USA E-mail: [email protected] Received 24 October 2014, revised 20 November 2014 Accepted for publication 25 November 2014 Published 15 December 2014 Abstract
Ways to increase the N´eel temperature TN in the magnetoelectric Fe2 TeO6 antiferromagnet are explored with the help of first-principles calculations. Substitution of larger ions like Zr or Hf for tellurium increases the superexchange angles. The compensating O vacancies tend to form bound complexes with Zr dopants, which do not degrade the electronic band gap. TN is estimated to increase by 15% at 12.5% Te→Zr substitution with such compensation. Substitution of N for O is favorable due to the decreased charge-transfer gap. The overall effect for N3− substitution compensated by O vacancies is estimated at 3–4% TN enhancement per 1% O→N substitution. A 1% compressive (0 0 1) epitaxial strain enhances TN by about 6%. Keywords: magnetoelectric antiferromagnet, first-principles calculations, superexchange (Some figures may appear in colour only in the online journal)
Spintronic device applications usually employ ferromagnets as active magnetic elements [1], but alternative schemes based on antiferromagnetic materials are also possible [2–7]. Antiferromagnets are used to provide an exchange bias [8] for pinned ferromagnetic layers in spin valves and tunnel junctions. Of particular interest are magnetoelectric [9, 10] antiferromagnets enabling voltage-controlled switching of exchange bias [11] with potential applications in nonvolatile memory [12, 13] and other devices. The lack of magnetoelectric antiferromagnets suitable for room-temperature operation motivated the search for alloying elements to increase the N´eel temperature TN of the ‘firstchoice’ magnetoelectric Cr2 O3 beyond its normal value of 307 K. A strong favorable effect of boron doping predicted theoretically for this material [14] was recently corroborated experimentally [15]. In this Communication we turn to Fe2 TeO6 , a magnetoelectric antiferromagnet with the trirutile lattice and TN of about 210 K [16]. The demonstration of electric-field control of spin polarization in the x-ray magnetic circular dichroic signal from Fe2 TeO6 surface [17] showed that this material is potentially suitable for switchable exchange bias 0953-8984/15/022203+05$33.00
applications. Our present goal is to develop strategies for increasing the N´eel temperature of Fe2 TeO6 . First-principles calculations were carried out using the projector augmented-wave (PAW) method [18] implemented in the Vienna ab initio simulation package (VASP) [19, 20]. The 3d states of Fe were treated within the LDA + U method [21]. First we address the magnetic properties of ideal Fe2 TeO6 . Its body-centered tetragonal crystal lattice can be represented as a Fe↑ Fe↓ Te [0 0 1] cation superlattice within the parent rutile lattice (figure 1(a)). Each Fe atom has one nearest Fe neighbor and four next-nearest Fe neighbors connected by superexchange paths shown in figure 1(b). Calculations with the constrained occupations of the Fe 3d states treated in the atomic limit give the values U = 6.0 eV and J = 0.9 eV. The full and partial densities of states (DOS) calculated with these parameters are plotted in figure 2. The top of the valence band is dominated by O 2p states, while empty Fe 3d states form the bottom of conduction band. The charge-transfer band gap is about 1.7 eV. The 3d states of Fe are half-filled, resulting in a high-spin state with the 4.3 µB local spin moment in good agreement with the experimental value of 4.2 µB [22]. 1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 022203
Fe spin moment. The calculated value U = 6 eV was used in further computations. We now turn to the possible strategies for increasing TN in Fe2 TeO6 , which is the main focus of this Communication. Since superexchange is maximized for half-filled 3d shells, substitution of other elements for Fe would be counterproductive. On the other hand, superexchange originates in the electronic processes that are fourth-order in the p–d overlap integrals tpd with energy denominators containing powers of the p–d charge-transfer gap [26]. Therefore, superexchange can be strengthened by increasing tpd or by decreasing . In the following we explore both of these options through suitable Te and O substitutions. To assess the effect of an impurity on TN , we use the approach of [14] and calculate the exchange energies for different Fe atoms in the supercell, defined as the energy cost of reversing the spin moment on a particular atom. It is then assumed that TN is proportional to the exchange energy averaged over all Fe atoms. The atomic positions are fixed at their optimized values for the antiferromagnetic state with the given impurities. Each Fe atom has one nearest Fe neighbor and four next-nearest neighbors (figure 1(b)). The J1 interaction is mediated by two O1 atoms in the TeO2 layer, and J2 by one O2 atom. The bond angles are 98◦ and 128◦ for Fe–O1 –Fe and Fe–O2 –Fe exchange paths, respectively. According to the Goodenough–Kanamori rules [27], a 180◦ bond angle is optimal for antiferromagnetic superexchange due to the maximization of the σ -bonding overlap integrals tpd . This is consistent with J2 being dominant. While for the 90◦ bond angle the σ -bonding superexchange is suppressed, there are several weaker superexchange mechanisms of different sign for two d 5 ions [27]. The fact that J1 is still strongly antiferromagnetic in spite of the bond angle being close to 90◦ suggests that the antiferromagnetic contribution from direct 3d–3d hopping also contributes to J1 . Since the Fe–O–Fe bond angles are far from optimal, our first strategy is to increase them by a suitable substitution. As indicated in figure 1(b), this can be achieved by substituting a larger ion for Te, which would push its O neighbors outward and thereby straighten the Fe–O–Fe bonds. Unfortunately, isoelectronic larger-ion substitutions for Te are unavailable; Mo and W are isoelectronic but we found they have little effect on the Fe–O–Fe angles and TN . We therefore consider groupIV (Zr, Hf) and group-V (Nb, Ta) substitutions. These defects should be compensated to preserve electric insulation. The substitution is implemented by replacing one of the eight Te atoms in the 72-atom supercell by the dopant (figure 1(b)). Let us focus on Zr (Hf is similar). The uncompensated Zr impurity introduces two holes: a mid-gap impurity level and a shallow donor level which blends with the valence band in our supercell, turning it metallic. Such shallow donor states are undesirable for applications because they easily degrade the electric insulation. We therefore consider a Zr4+ defect state by adding a positive charge 2e as homogeneous background representing the presence of yet unspecified compensating defects. The extra charge fills the hole states introduced by Zr. As a result of the ensuing
Figure 1. (a) Tetragonal unit cell of Fe2 TeO6 . (b) Side view of the 2 × 2 × 1 supercell. Arrows: superexchange paths for J1 and J2 . Dashed lines: lattice distortion near a large impurity like Zr on the Te site (sketch).
Figure 2. Density of states (DOS) in Fe2 TeO6 . Gray line: total DOS per formula unit. Blue, red and green lines show partial DOS for Fe, O (times 6) and Te, respectively.
The large deviation from the 5 µB ionic limit is due to the hybridization of the Fe 3d and O 2p states, which is evident in the partial DOS. To evaluate the exchange interaction parameters and to study the effects of chemical substitution, we set up a 72-atom 2 × 2 × 1 supercell whose side view is shown in figure 1(b). The Heisenberg exchange parameters Jn are fitted to the calculated total energies of 14 collinear spin configurations in this supercell. (The Jn are normalized so that an n-shell pair of parallel spins contributes (5/2)2 Jn to the total energy.) The N´eel temperature is estimated using the spin-5/2 pair cluster approximation [23], which takes into account the pairwise spin correlations in the spirit of the Bethe–Peierls approximation; see [24] for its application to Cr2 O3 . The key properties are listed in table 1 for several values of U . The volume and the bond angles (not listed) are insensitive to U ; the volume is underestimated by about 3% compared to experiment [25]. The dominant exchange couplings J1 and J2 correspond to Fe atoms directly linked by O. At U = 6 eV we find TN = 281 K, which exceeds the experimental value [16] by 34%. TN decreases quickly with increasing U due to the suppression of the p–d hybridization, which also increases the 2
J. Phys.: Condens. Matter 27 (2015) 022203
Table 1. Exchange parameters Jn (meV) as a function of U (eV) at J = 0.9 eV. CV: take-one-out cross-validation score (meV) for each fit. The atomic volume (Å3 /atom), the Fe spin moment µFe (µB ), the band gap g (eV), and the calculated N´eel temperature TN (K) are also listed.
U
µFe
g
J1
J2
J3
J4
J5
CV
TN
4.0 5.0 6.0 7.0 8.0
10.42 10.41 10.38 10.34 10.27
4.11 4.21 4.30 4.39 4.50
1.42 1.56 1.69 1.87 2.03
3.41 2.82 2.19 1.68 1.29
4.66 3.82 3.16 2.58 2.09
0.81 0.63 0.51 0.41 0.32
0.72 0.52 0.42 0.34 0.26
0.31 0.21 0.15 0.11 0.08
0.3 0.3 0.2 0.5 0.1
379 329 281 234 191
Table 2. Fe–O bond lengths (Å) and Fe–O–Fe bond angles near the impurity in the Fe16 XTe7 O48 supercell. The last column (η) shows the estimated increase of TN .
X Te Mo Nb5+ Zr4+ W Ta5+ Hf4+
Fe–O1 1.97 1.96 1.92 1.88 1.96 1.92 1.88
Fe–O2 1.97, 1.99 1.97, 2.00 1.92, 1.96 1.91, 1.91 1.96, 1.99 1.92, 1.96 1.88, 1,91
Fe–O1 –Fe ◦
98.3 97.9◦ 100.2◦ 103.7◦ 97.9◦ 100.2◦ 102.7◦
Fe–O2 –Fe ◦
128.3 126.9◦ 130.6◦ 136.3◦ 127.2◦ 130.8◦ 134.8◦
η, % 0 3.6 10.7 23.8 −3.3 10.2 21.4
structural relaxation the mid-gap state moves down and blends with the valence band. A full insulating band gap with no impurity levels is thereby restored. This favorable property is shared by Hf4+ , Nb5+ , and Ta5+ impurities. Table 2 shows that the Fe–O bond lengths decrease and the Fe–O–Fe angles increase near the Zr4+ impurity, leading to a 24% increase in TN . The Hf4+ impurity has a similar but slightly smaller effect. Group-V impurities (Nb and Ta) are about twice less efficient, while group-VI Mo and W have nearly no effect on TN . Oxygen vacancies (VO ) can serve as compensating defects for group-IV and group-V impurities. In particular, an O vacancy can donate two electrons to compensate the two holes introduced by a Zr impurity. Figure 3 shows the total energy of the 72-atom supercell with one Zr (or Hf) impurity and one O vacancy as a function of the distance between the two defects. The lowest energy corresponds to the O vacancy located at the O2 site next to Zr. Transferring the vacancy to the O1 site next to Zr costs almost 0.4 eV, and moving it further away at least 0.7 eV. Clearly, O vacancies tend to associate in strongly bound complexes with oppositely-charged Zr impurities. The free energy of vacancy formation can be controlled by the partial pressure of oxygen gas. It is seen from figure 3 that there is a wide range of oxygen chemical potential such that the formation of a vacancy in the O2 site near Zr lowers the free energy, while unbound O vacancies are unfavorable. Annealing at the corresponding oxygen pressure can therefore be used to promote the binding of one O vacancy to each Zr impurity while keeping the unbound vacancy concentration low. The DOS of a supercell with one stable Zr–VO2 complex (not shown) has no impurity levels in the band gap. The same applies to the nearest-neighbor Zr–VO1 complex with the nextlowest energy. Therefore, such complexes are not detrimental to the electric insulation of the material. An undesirable side effect of the O vacancies is the deletion of the superexchange paths mediated by the missing
Figure 3. Total energy of a supercell with one Zr (or Hf) impurity and one O vacancy as a function of the distance between the two defects. Blue line: Zr; red: Hf. The ground state energy is taken as zero.
oxygen atoms. The net effect on the TN is, however, still positive: the replacement of one Te by a stable Zr–VO2 complex in the 72-atom supercell leads to a 15% enhancement of TN . Similar conclusions apply to Hf impurities. Another way to increase TN is to add anion impurities with a smaller 2p binding energy and hence a smaller p–d chargetransfer energy. In order to preserve the antiferromagnetic superexchange, the anion 2p states should remain filled. Thus, the anion 2p states should be raised but not too much, which suggests nitrogen substitution. Calculations for the 72-atom supercell show that the substitution of N at the O2 site has a lower energy compared to the O1 site by 132 meV and is therefore more favorable. The following results are presented for this case, but in all important aspects they are similar for O1 substitution. The DOS for N substitution at O2 is shown in figure 4 for two charge states: N2− (neutral supercell) and N3− (extra charge e added as positive background). A N2− dopant introduces an empty mid-gap impurity level, which is strongly localized on the N atom. There is also an occupied level close to the valence band maximum. The N atom has a large local spin moment of 0.5 µB due to the localized spin-polarized hole. The addition of one electron (N3− charge state) splits off two more localized states from the valence band. As seen from figure 4, the excitation gap is significantly larger in this charge state, which should lead to better electric insulation, although 3
J. Phys.: Condens. Matter 27 (2015) 022203
Figure 5. (a) Fe–O bond lengths and (b) J1 , J2 and TN as a function of epitaxial strain.
Figure 4. DOS for the N dopant at the O2 site in Fe2 TeO6 : neutral
state (upper panel) and the state with one extra electron (lower panel). Gray line: total DOS; green line: partial DOS for N; blue and red lines: partial DOS for the two Fe neighbors of N. All partial DOS are scaled by factor 2.
the empty impurity orbital has a weaker (p–d)π bonding character. Using the averaged exchange energies, we estimate the TN increases by 2–3% or 5–6% for substitution at a 1% N–O ratio in the N2− or N3− charge state, respectively. Clearly, the N3− state is preferable, but in practice these defects need to be compensated. Since one O vacancy is required to compensate two N3− dopants, we refrain from studing the possible defect complexes here. If the negative effect of O vacancies on TN is similar to the compensated Zr case, the overall effect of vacancy-compensated N3− should be reduced to 3–4% at the 1% N–O ratio. We now consider the effect on TN of mechanical strain uαβ , which changes the overlap integrals tpd through changes in bond lengths and geometry. Strain-induced changes in TN are described by a second-rank tensor, which in a tetragonal crystal has two independent components: δTN = ηz uzz +η⊥ (uxx +uyy ). In order to determine ηz and η⊥ , we consider two stress modes preserving the tetragonal symmetry: epitaxial stress (vanishing σzz stress component) and uniaxial stress (vanishing σxx and σyy ). All atomic positions are relaxed along with the lattice parameter in the direction of vanishing stress. The uniaxial stress has almost no effect on TN , so we only display the epitaxial strain results in figure 5. As expected for this strain mode, the in-plane Fe–O2 bonds expand or contract in proportion to the in-plane strain, while other Fe–O bond lengths are almost unchanged. Under epitaxial compression, the Fe–O1 –Fe bond angle increases due to the increase in the c parameter, and therefore J1 increases. Although the Fe–O2 – Fe angle decreases at a rate of 2◦ per 1% compressive strain,
Table 3. Fe–N–Fe bond angle, Fe–N bond lengths di (Å), and exchange energies (Ei , meV) for the two Fe neighbors of the N dopant, compared with the undoped case.
Dopant O N2− N3−
Bond angle ◦
128.3 131.2◦ 130.1◦
d1
d2
E1
E2
1.97 1.97 1.90
1.99 2.00 1.88
164 212 310
164 264 322
it is still reduced by about 50% compared to the bulk band gap. The raising and complete filling of the N 2p states can both be expected to strengthen the antiferromagnetic superexchange interaction; this is confirmed by the data in table 3. The exchange energies of the two Fe neighbors of N are larger for the N2− state compared to the undoped case, and larger still for the N3− impurity. Although the N3− impurity shortens the bonds and slightly increases the bond angle, these changes have little effect on the exchange energies. Indeed, we found the exchange energies E1 and E2 become only 3–4% larger compared to bulk Fe2 TeO6 if N is replaced back by O while fixing the atomic positions. Thus, the enhanced exchange coupling is due to the reduced chargetransfer energy. Note that the exchange energies are already enhanced in the N2− charge state, even though a partially filled 2p orbital should make a ferromagnetic contribution to exchange interaction. This is explained by the fact that 4
J. Phys.: Condens. Matter 27 (2015) 022203
J2 also increases due to the contraction of the in-plane Fe–O2 bonds. The dependence of J1 on strain is strongly nonlinear on the tensile side, with J1 passing through a minimum at 3% strain. This is likely due to the antiferromagnetic contribution from direct 3d–3d hopping, which is promoted by shorter nearest-neighbor Fe–Fe distance. In the linear region TN increases at a rate of 6% per 1% epitaxial compression and is almost constant under uniaxial stress. In terms of pressure, dTN /dpa = dTN /dpb = 0.25 K/kbar, dTN /dpc < 0.01 K/kbar. The tensor components defined above are ηz = −700 K and η⊥ = −1160 K. In conclusion, the N´eel temperature of Fe2 TeO6 can be enhanced while maintaining electric insulation by (1) substitution of larger ions like Zr for Te with full compensation by O vacancies by tuning the oxygen gas pressure, (2) substitution of N for O, perhaps also compensated by O vacancies, and (3) compressive (0 0 1) epitaxial strain.
[4] Barthem V M T S, Colin C V, Mayaffre H, Julien M-H and Givord D 2013 Nat. Commun. 4 2892 [5] Wadley P et al 2013 Nat. Commun. 4 2322 [6] Wu H-C, Liao Z-M, Sofin R G S, Feng G, Ma X-M, Shick A B, Mryasov O N and Shvets I V 2012 Adv. Mater. 24 6374 [7] Park B G et al 2011 Nat. Mater. 10 347 [8] Nogu´es J et al 2005 Phys. Rep. 422 65 [9] Landau L D and Lifshitz E M 1960 Electrodynamics of Continuous Media (Oxford: Pergamon) [10] Fiebig M 2005 J. Phys. D 38 R123 [11] He X et al 2010 Nat. Mater. 9 579 [12] Binek Ch and Doudin B 2005 J. Phys.: Condens. Matter 17 L39 [13] Bibes M and Barth´el´emy A 2008 Nat. Mater. 7 425 [14] Mu S, Wysocki A L and Belashchenko K D 2013 Phys. Rev. B 87 054435 [15] Street M et al 2014 Appl. Phys. Lett. 104 222402 [16] Buksphan S, Fischer E and Hornreich R 1972 Solid State Commun. 10 657 [17] Wang J et al 2014 J. Phys.: Condens. Matter 26 055012 [18] Bl¨ochl P E 1994 Phys. Rev. B 50 17953 [19] Kresse G and Hafner J 1993 Phys. Rev. B 48 13115 [20] Kresse G and Furthm¨uller J 1996 Phys. Rev. B 54 11169 [21] Liechtenstein A I, Anisimov V I and Zaanen J 1995 Phys. Rev. B 52 R5467 [22] Kunnmann W, La Placa S, Corliss L M and Hastings J M 1968 J. Phys. Chem. Solids 29 1359 [23] Vaks V G and Zein N E 1975 Sov. Phys.—JETP 40 537 [24] Shi S, Wysocki A L and Belashchenko K D 2009 Phys. Rev. B 79 104404 [25] Krishnan K, Mudher K D S, Rao G A R and Venugopal V 2001 J. Alloys Compounds 316 264 [26] Maekawa S, Tohyama T, Barnes S, Ishihara S, Koshibae W and Khaliullin G 2004 Physics of Transition Metal Oxides (Springer Series in Solid State Sciences) vol 144 (Berlin: Springer) [27] Goodenough J B 1958 J. Phys. Chem. Solids 6 287 Kanamori J 1959 J. Phys. Chem. Solids 10 87
Acknowledgments
This work was supported by the Center for NanoFerroic Devices (CNFD) and the Nanoelectronics Research Initiative (NRI) and performed utilizing the Holland Computing Center of the University of Nebraska. References ˇ c I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. [1] Zuti´ 76 323 [2] N´un˜ ez A S, Duine R A, Haney P and MacDonald A H 2006 Phys. Rev. B 73 214426 [3] Shick A B, Khmelevskyi S, Mryasov O N, Wunderlich J and Jungwirth T 2010 Phys. Rev. B 81 212409
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Strategies for increasing the Néel temperature of magnetoelectric Fe2TeO6
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 022203 (http://iopscience.iop.org/0953-8984/27/2/022203) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 155.246.15.35 This content was downloaded on 21/02/2015 at 11:26
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 022203 (5pp)
doi:10.1088/0953-8984/27/2/022203
Fast Track Communication
´ Strategies for increasing the Neel temperature of magnetoelectric Fe2TeO6 Sai Mu and Kirill D Belashchenko Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, Lincoln, NE 68588, USA E-mail: [email protected] Received 24 October 2014, revised 20 November 2014 Accepted for publication 25 November 2014 Published 15 December 2014 Abstract
Ways to increase the N´eel temperature TN in the magnetoelectric Fe2 TeO6 antiferromagnet are explored with the help of first-principles calculations. Substitution of larger ions like Zr or Hf for tellurium increases the superexchange angles. The compensating O vacancies tend to form bound complexes with Zr dopants, which do not degrade the electronic band gap. TN is estimated to increase by 15% at 12.5% Te→Zr substitution with such compensation. Substitution of N for O is favorable due to the decreased charge-transfer gap. The overall effect for N3− substitution compensated by O vacancies is estimated at 3–4% TN enhancement per 1% O→N substitution. A 1% compressive (0 0 1) epitaxial strain enhances TN by about 6%. Keywords: magnetoelectric antiferromagnet, first-principles calculations, superexchange (Some figures may appear in colour only in the online journal)
Spintronic device applications usually employ ferromagnets as active magnetic elements [1], but alternative schemes based on antiferromagnetic materials are also possible [2–7]. Antiferromagnets are used to provide an exchange bias [8] for pinned ferromagnetic layers in spin valves and tunnel junctions. Of particular interest are magnetoelectric [9, 10] antiferromagnets enabling voltage-controlled switching of exchange bias [11] with potential applications in nonvolatile memory [12, 13] and other devices. The lack of magnetoelectric antiferromagnets suitable for room-temperature operation motivated the search for alloying elements to increase the N´eel temperature TN of the ‘firstchoice’ magnetoelectric Cr2 O3 beyond its normal value of 307 K. A strong favorable effect of boron doping predicted theoretically for this material [14] was recently corroborated experimentally [15]. In this Communication we turn to Fe2 TeO6 , a magnetoelectric antiferromagnet with the trirutile lattice and TN of about 210 K [16]. The demonstration of electric-field control of spin polarization in the x-ray magnetic circular dichroic signal from Fe2 TeO6 surface [17] showed that this material is potentially suitable for switchable exchange bias 0953-8984/15/022203+05$33.00
applications. Our present goal is to develop strategies for increasing the N´eel temperature of Fe2 TeO6 . First-principles calculations were carried out using the projector augmented-wave (PAW) method [18] implemented in the Vienna ab initio simulation package (VASP) [19, 20]. The 3d states of Fe were treated within the LDA + U method [21]. First we address the magnetic properties of ideal Fe2 TeO6 . Its body-centered tetragonal crystal lattice can be represented as a Fe↑ Fe↓ Te [0 0 1] cation superlattice within the parent rutile lattice (figure 1(a)). Each Fe atom has one nearest Fe neighbor and four next-nearest Fe neighbors connected by superexchange paths shown in figure 1(b). Calculations with the constrained occupations of the Fe 3d states treated in the atomic limit give the values U = 6.0 eV and J = 0.9 eV. The full and partial densities of states (DOS) calculated with these parameters are plotted in figure 2. The top of the valence band is dominated by O 2p states, while empty Fe 3d states form the bottom of conduction band. The charge-transfer band gap is about 1.7 eV. The 3d states of Fe are half-filled, resulting in a high-spin state with the 4.3 µB local spin moment in good agreement with the experimental value of 4.2 µB [22]. 1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 022203
Fe spin moment. The calculated value U = 6 eV was used in further computations. We now turn to the possible strategies for increasing TN in Fe2 TeO6 , which is the main focus of this Communication. Since superexchange is maximized for half-filled 3d shells, substitution of other elements for Fe would be counterproductive. On the other hand, superexchange originates in the electronic processes that are fourth-order in the p–d overlap integrals tpd with energy denominators containing powers of the p–d charge-transfer gap [26]. Therefore, superexchange can be strengthened by increasing tpd or by decreasing . In the following we explore both of these options through suitable Te and O substitutions. To assess the effect of an impurity on TN , we use the approach of [14] and calculate the exchange energies for different Fe atoms in the supercell, defined as the energy cost of reversing the spin moment on a particular atom. It is then assumed that TN is proportional to the exchange energy averaged over all Fe atoms. The atomic positions are fixed at their optimized values for the antiferromagnetic state with the given impurities. Each Fe atom has one nearest Fe neighbor and four next-nearest neighbors (figure 1(b)). The J1 interaction is mediated by two O1 atoms in the TeO2 layer, and J2 by one O2 atom. The bond angles are 98◦ and 128◦ for Fe–O1 –Fe and Fe–O2 –Fe exchange paths, respectively. According to the Goodenough–Kanamori rules [27], a 180◦ bond angle is optimal for antiferromagnetic superexchange due to the maximization of the σ -bonding overlap integrals tpd . This is consistent with J2 being dominant. While for the 90◦ bond angle the σ -bonding superexchange is suppressed, there are several weaker superexchange mechanisms of different sign for two d 5 ions [27]. The fact that J1 is still strongly antiferromagnetic in spite of the bond angle being close to 90◦ suggests that the antiferromagnetic contribution from direct 3d–3d hopping also contributes to J1 . Since the Fe–O–Fe bond angles are far from optimal, our first strategy is to increase them by a suitable substitution. As indicated in figure 1(b), this can be achieved by substituting a larger ion for Te, which would push its O neighbors outward and thereby straighten the Fe–O–Fe bonds. Unfortunately, isoelectronic larger-ion substitutions for Te are unavailable; Mo and W are isoelectronic but we found they have little effect on the Fe–O–Fe angles and TN . We therefore consider groupIV (Zr, Hf) and group-V (Nb, Ta) substitutions. These defects should be compensated to preserve electric insulation. The substitution is implemented by replacing one of the eight Te atoms in the 72-atom supercell by the dopant (figure 1(b)). Let us focus on Zr (Hf is similar). The uncompensated Zr impurity introduces two holes: a mid-gap impurity level and a shallow donor level which blends with the valence band in our supercell, turning it metallic. Such shallow donor states are undesirable for applications because they easily degrade the electric insulation. We therefore consider a Zr4+ defect state by adding a positive charge 2e as homogeneous background representing the presence of yet unspecified compensating defects. The extra charge fills the hole states introduced by Zr. As a result of the ensuing
Figure 1. (a) Tetragonal unit cell of Fe2 TeO6 . (b) Side view of the 2 × 2 × 1 supercell. Arrows: superexchange paths for J1 and J2 . Dashed lines: lattice distortion near a large impurity like Zr on the Te site (sketch).
Figure 2. Density of states (DOS) in Fe2 TeO6 . Gray line: total DOS per formula unit. Blue, red and green lines show partial DOS for Fe, O (times 6) and Te, respectively.
The large deviation from the 5 µB ionic limit is due to the hybridization of the Fe 3d and O 2p states, which is evident in the partial DOS. To evaluate the exchange interaction parameters and to study the effects of chemical substitution, we set up a 72-atom 2 × 2 × 1 supercell whose side view is shown in figure 1(b). The Heisenberg exchange parameters Jn are fitted to the calculated total energies of 14 collinear spin configurations in this supercell. (The Jn are normalized so that an n-shell pair of parallel spins contributes (5/2)2 Jn to the total energy.) The N´eel temperature is estimated using the spin-5/2 pair cluster approximation [23], which takes into account the pairwise spin correlations in the spirit of the Bethe–Peierls approximation; see [24] for its application to Cr2 O3 . The key properties are listed in table 1 for several values of U . The volume and the bond angles (not listed) are insensitive to U ; the volume is underestimated by about 3% compared to experiment [25]. The dominant exchange couplings J1 and J2 correspond to Fe atoms directly linked by O. At U = 6 eV we find TN = 281 K, which exceeds the experimental value [16] by 34%. TN decreases quickly with increasing U due to the suppression of the p–d hybridization, which also increases the 2
J. Phys.: Condens. Matter 27 (2015) 022203
Table 1. Exchange parameters Jn (meV) as a function of U (eV) at J = 0.9 eV. CV: take-one-out cross-validation score (meV) for each fit. The atomic volume (Å3 /atom), the Fe spin moment µFe (µB ), the band gap g (eV), and the calculated N´eel temperature TN (K) are also listed.
U
µFe
g
J1
J2
J3
J4
J5
CV
TN
4.0 5.0 6.0 7.0 8.0
10.42 10.41 10.38 10.34 10.27
4.11 4.21 4.30 4.39 4.50
1.42 1.56 1.69 1.87 2.03
3.41 2.82 2.19 1.68 1.29
4.66 3.82 3.16 2.58 2.09
0.81 0.63 0.51 0.41 0.32
0.72 0.52 0.42 0.34 0.26
0.31 0.21 0.15 0.11 0.08
0.3 0.3 0.2 0.5 0.1
379 329 281 234 191
Table 2. Fe–O bond lengths (Å) and Fe–O–Fe bond angles near the impurity in the Fe16 XTe7 O48 supercell. The last column (η) shows the estimated increase of TN .
X Te Mo Nb5+ Zr4+ W Ta5+ Hf4+
Fe–O1 1.97 1.96 1.92 1.88 1.96 1.92 1.88
Fe–O2 1.97, 1.99 1.97, 2.00 1.92, 1.96 1.91, 1.91 1.96, 1.99 1.92, 1.96 1.88, 1,91
Fe–O1 –Fe ◦
98.3 97.9◦ 100.2◦ 103.7◦ 97.9◦ 100.2◦ 102.7◦
Fe–O2 –Fe ◦
128.3 126.9◦ 130.6◦ 136.3◦ 127.2◦ 130.8◦ 134.8◦
η, % 0 3.6 10.7 23.8 −3.3 10.2 21.4
structural relaxation the mid-gap state moves down and blends with the valence band. A full insulating band gap with no impurity levels is thereby restored. This favorable property is shared by Hf4+ , Nb5+ , and Ta5+ impurities. Table 2 shows that the Fe–O bond lengths decrease and the Fe–O–Fe angles increase near the Zr4+ impurity, leading to a 24% increase in TN . The Hf4+ impurity has a similar but slightly smaller effect. Group-V impurities (Nb and Ta) are about twice less efficient, while group-VI Mo and W have nearly no effect on TN . Oxygen vacancies (VO ) can serve as compensating defects for group-IV and group-V impurities. In particular, an O vacancy can donate two electrons to compensate the two holes introduced by a Zr impurity. Figure 3 shows the total energy of the 72-atom supercell with one Zr (or Hf) impurity and one O vacancy as a function of the distance between the two defects. The lowest energy corresponds to the O vacancy located at the O2 site next to Zr. Transferring the vacancy to the O1 site next to Zr costs almost 0.4 eV, and moving it further away at least 0.7 eV. Clearly, O vacancies tend to associate in strongly bound complexes with oppositely-charged Zr impurities. The free energy of vacancy formation can be controlled by the partial pressure of oxygen gas. It is seen from figure 3 that there is a wide range of oxygen chemical potential such that the formation of a vacancy in the O2 site near Zr lowers the free energy, while unbound O vacancies are unfavorable. Annealing at the corresponding oxygen pressure can therefore be used to promote the binding of one O vacancy to each Zr impurity while keeping the unbound vacancy concentration low. The DOS of a supercell with one stable Zr–VO2 complex (not shown) has no impurity levels in the band gap. The same applies to the nearest-neighbor Zr–VO1 complex with the nextlowest energy. Therefore, such complexes are not detrimental to the electric insulation of the material. An undesirable side effect of the O vacancies is the deletion of the superexchange paths mediated by the missing
Figure 3. Total energy of a supercell with one Zr (or Hf) impurity and one O vacancy as a function of the distance between the two defects. Blue line: Zr; red: Hf. The ground state energy is taken as zero.
oxygen atoms. The net effect on the TN is, however, still positive: the replacement of one Te by a stable Zr–VO2 complex in the 72-atom supercell leads to a 15% enhancement of TN . Similar conclusions apply to Hf impurities. Another way to increase TN is to add anion impurities with a smaller 2p binding energy and hence a smaller p–d chargetransfer energy. In order to preserve the antiferromagnetic superexchange, the anion 2p states should remain filled. Thus, the anion 2p states should be raised but not too much, which suggests nitrogen substitution. Calculations for the 72-atom supercell show that the substitution of N at the O2 site has a lower energy compared to the O1 site by 132 meV and is therefore more favorable. The following results are presented for this case, but in all important aspects they are similar for O1 substitution. The DOS for N substitution at O2 is shown in figure 4 for two charge states: N2− (neutral supercell) and N3− (extra charge e added as positive background). A N2− dopant introduces an empty mid-gap impurity level, which is strongly localized on the N atom. There is also an occupied level close to the valence band maximum. The N atom has a large local spin moment of 0.5 µB due to the localized spin-polarized hole. The addition of one electron (N3− charge state) splits off two more localized states from the valence band. As seen from figure 4, the excitation gap is significantly larger in this charge state, which should lead to better electric insulation, although 3
J. Phys.: Condens. Matter 27 (2015) 022203
Figure 5. (a) Fe–O bond lengths and (b) J1 , J2 and TN as a function of epitaxial strain.
Figure 4. DOS for the N dopant at the O2 site in Fe2 TeO6 : neutral
state (upper panel) and the state with one extra electron (lower panel). Gray line: total DOS; green line: partial DOS for N; blue and red lines: partial DOS for the two Fe neighbors of N. All partial DOS are scaled by factor 2.
the empty impurity orbital has a weaker (p–d)π bonding character. Using the averaged exchange energies, we estimate the TN increases by 2–3% or 5–6% for substitution at a 1% N–O ratio in the N2− or N3− charge state, respectively. Clearly, the N3− state is preferable, but in practice these defects need to be compensated. Since one O vacancy is required to compensate two N3− dopants, we refrain from studing the possible defect complexes here. If the negative effect of O vacancies on TN is similar to the compensated Zr case, the overall effect of vacancy-compensated N3− should be reduced to 3–4% at the 1% N–O ratio. We now consider the effect on TN of mechanical strain uαβ , which changes the overlap integrals tpd through changes in bond lengths and geometry. Strain-induced changes in TN are described by a second-rank tensor, which in a tetragonal crystal has two independent components: δTN = ηz uzz +η⊥ (uxx +uyy ). In order to determine ηz and η⊥ , we consider two stress modes preserving the tetragonal symmetry: epitaxial stress (vanishing σzz stress component) and uniaxial stress (vanishing σxx and σyy ). All atomic positions are relaxed along with the lattice parameter in the direction of vanishing stress. The uniaxial stress has almost no effect on TN , so we only display the epitaxial strain results in figure 5. As expected for this strain mode, the in-plane Fe–O2 bonds expand or contract in proportion to the in-plane strain, while other Fe–O bond lengths are almost unchanged. Under epitaxial compression, the Fe–O1 –Fe bond angle increases due to the increase in the c parameter, and therefore J1 increases. Although the Fe–O2 – Fe angle decreases at a rate of 2◦ per 1% compressive strain,
Table 3. Fe–N–Fe bond angle, Fe–N bond lengths di (Å), and exchange energies (Ei , meV) for the two Fe neighbors of the N dopant, compared with the undoped case.
Dopant O N2− N3−
Bond angle ◦
128.3 131.2◦ 130.1◦
d1
d2
E1
E2
1.97 1.97 1.90
1.99 2.00 1.88
164 212 310
164 264 322
it is still reduced by about 50% compared to the bulk band gap. The raising and complete filling of the N 2p states can both be expected to strengthen the antiferromagnetic superexchange interaction; this is confirmed by the data in table 3. The exchange energies of the two Fe neighbors of N are larger for the N2− state compared to the undoped case, and larger still for the N3− impurity. Although the N3− impurity shortens the bonds and slightly increases the bond angle, these changes have little effect on the exchange energies. Indeed, we found the exchange energies E1 and E2 become only 3–4% larger compared to bulk Fe2 TeO6 if N is replaced back by O while fixing the atomic positions. Thus, the enhanced exchange coupling is due to the reduced chargetransfer energy. Note that the exchange energies are already enhanced in the N2− charge state, even though a partially filled 2p orbital should make a ferromagnetic contribution to exchange interaction. This is explained by the fact that 4
J. Phys.: Condens. Matter 27 (2015) 022203
J2 also increases due to the contraction of the in-plane Fe–O2 bonds. The dependence of J1 on strain is strongly nonlinear on the tensile side, with J1 passing through a minimum at 3% strain. This is likely due to the antiferromagnetic contribution from direct 3d–3d hopping, which is promoted by shorter nearest-neighbor Fe–Fe distance. In the linear region TN increases at a rate of 6% per 1% epitaxial compression and is almost constant under uniaxial stress. In terms of pressure, dTN /dpa = dTN /dpb = 0.25 K/kbar, dTN /dpc < 0.01 K/kbar. The tensor components defined above are ηz = −700 K and η⊥ = −1160 K. In conclusion, the N´eel temperature of Fe2 TeO6 can be enhanced while maintaining electric insulation by (1) substitution of larger ions like Zr for Te with full compensation by O vacancies by tuning the oxygen gas pressure, (2) substitution of N for O, perhaps also compensated by O vacancies, and (3) compressive (0 0 1) epitaxial strain.
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Acknowledgments
This work was supported by the Center for NanoFerroic Devices (CNFD) and the Nanoelectronics Research Initiative (NRI) and performed utilizing the Holland Computing Center of the University of Nebraska. References ˇ c I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. [1] Zuti´ 76 323 [2] N´un˜ ez A S, Duine R A, Haney P and MacDonald A H 2006 Phys. Rev. B 73 214426 [3] Shick A B, Khmelevskyi S, Mryasov O N, Wunderlich J and Jungwirth T 2010 Phys. Rev. B 81 212409
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